Consider the 50.0-kg mountain climber in
the figure to the right. Find (a) the tension
in the rope and (b) the force that the
mountain climber must exert with her feet
on the vertical rock face to remain
stationary. Assume that the force is exerted
parallel to her legs. Also, assume negligible
force exerted by her arms. (c) What is the minimum coefficient of friction between her shoes and the cliff.
I have managed to find part a using mgsin/cos(theta) the square root of those products squared. I ended up with 490N here.
Why does this process not work for the second force with the legs against the wall? And how do you find the coefficient from there?
All I see to the right is an ad.
To find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face, you can use Newton's laws of motion. Let's break down each part of the problem.
(a) Tension in the rope:
First, consider the forces acting on the mountain climber. We have the weight acting downwards, denoted by mg, where m is the mass of the climber (50.0 kg), and g is the acceleration due to gravity (9.8 m/s^2). We also have the tension force in the rope, pulling upwards.
When the mountain climber is in equilibrium, the sum of the forces in the vertical direction (upward forces equal downward forces) is equal to zero.
So, we have:
Tension - mg × cos(θ) = 0
Rearranging the equation, we find:
Tension = mg × cos(θ)
Using the given values, we get:
Tension = 50.0 kg × 9.8 m/s^2 × cos(θ)
(b) Force exerted by the feet on the vertical rock face:
Next, let's consider the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. This force is perpendicular to the surface of the cliff and counteracts the horizontal component of the tension force.
When the mountain climber is in equilibrium, the sum of the forces in the horizontal direction (leftward forces equal rightward forces) is equal to zero.
So, we have:
Force with feet - mg × sin(θ) = 0
Rearranging the equation, we find:
Force with feet = mg × sin(θ)
Using the given values, we get:
Force with feet = 50.0 kg × 9.8 m/s^2 × sin(θ)
(c) Minimum coefficient of friction:
To find the minimum coefficient of friction between her shoes and the cliff, we need to consider the maximum force of friction that can be exerted by the cliff.
The maximum force of friction is given by:
Friction = coefficient of friction × normal force
In this case, the normal force is equal to the force with feet, since the mountain climber is stationary in the vertical direction. So, we can write:
Friction = coefficient of friction × mg × sin(θ)
The minimum coefficient of friction is the coefficient value that allows the friction force to equal the force required with the feet to remain stationary. Therefore,
coefficient of friction = (mg × sin(θ)) / (mg × cos(θ))
The masses (m) and gravity (g) cancel out, simplifying the equation to:
coefficient of friction = tan(θ)
Therefore, the minimum coefficient of friction between her shoes and the cliff is simply the tangent of the angle θ.
To summarize,
(a) Tension in the rope: Tension = 50.0 kg × 9.8 m/s^2 × cos(θ)
(b) Force with feet: Force with feet = 50.0 kg × 9.8 m/s^2 × sin(θ)
(c) Minimum coefficient of friction: coefficient of friction = tan(θ)